To find an equivalent equation for [tex]\(\log_x(r) = -n\)[/tex], let's proceed step-by-step using the properties of logarithms and exponents.
1. Rewrite the logarithmic equation using the definition of a logarithm:
[tex]\[\log_x(r) = -n\][/tex]
The logarithm [tex]\(\log_x(r) = -n\)[/tex] can be interpreted as: "x raised to the power of [tex]\(-n\)[/tex] equals [tex]\(r\)[/tex]." In mathematical terms, this can be written as:
[tex]\[x^{-n} = r\][/tex]
2. Express the exponent as a reciprocal:
Recall that a negative exponent signifies a reciprocal. So, [tex]\(x^{-n}\)[/tex] can be rewritten as:
[tex]\[x^{-n} = \frac{1}{x^n}\][/tex]
3. Substitute back into the equation:
Now, substitute [tex]\(\frac{1}{x^n}\)[/tex] in place of [tex]\(x^{-n}\)[/tex]:
[tex]\[\frac{1}{x^n} = r\][/tex]
Thus, the equation equivalent to [tex]\(\log_x(r) = -n\)[/tex] is:
[tex]\[\boxed{r = \frac{1}{x^n}}\][/tex]
